These practice problems are not in the text book:

- Generate the first 11 rows of Pascals Triangle (stop when you’ve reached C(10,10)).
- Use your answer to number one to find C(8,6) (do not use the formula $latex \binom{8}{6}$).
- Check that the sum of all of the entries in the last row in your triangle is 2^10. What are you counting when you add the entries in a row of Pascal’s Triangle? (Solution: When you add all of the entries of the last row in your triangle you’re counting all of the subsets of the set {1,2,3,…,10}.)
- Verify that the number of all subsets of the set {1,2,3,4} is 2^4 either by using Pascal’s Triangle or by writing down all possible subsets.
- Use multiplication to count all of the subsets of the set {1,2,3,4}. (Hint: for each number between 1 and 4, there are two options: either the number is
**I**n the subset or it is**O**ut. For each subset, write down a corresponding sequence of**I**‘s and**O**‘s. Now the problem looks just like counting all possible outcomes when we toss a coin 4 times.) - Verify that the
*alternating*sum of the last row from your triangle is 0. (The*alternating*sum means that we alternate between adding and subtracting. For example, the alternating sum of the 4-th row in Pascal’s Triangle is 1-3+3-1.)

The follow problems come from Section 5.6 of your text book:

- 1,2,3,4,9,11,13,15 (For 9-15 odd, shortest possible route means that only
**S**outh and**E**ast steps are allowed.) - Solutions:
- 2 Part a: 2^9, Part b: C(9,2)
- 4 Part a: C(6,0)+C(6,1)+C(6,2), Part b: 2^6-22 (Hint: Think about our solution to number 1 in Quiz 7),

Here are selected solutions for problems not in your text book:

**Here’s a link to download: Selected solutions**